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is the general solution of equation (2). Now let 6 = 0 when t = 0, then c2 = — Ci. Therefore
6 = Atf~ at(eVa*"bn — <3~ v/a" ~'''"'),             (XXI)
where Ai = aci. There are three special cases which must be discussed separately.
Case I. Let a2 = 62, then 0 = 0 for all values of the time. Therefore this is a case of no motion.
Case II. Let a2 > 62, then Va2 — b'2 is real. Denoting this radical by c we have
The character of the motion is brought out by the graph of equation (XXII), Fig. 145. The graph is easily obtained by drawing the dotted curves, which are _ plotted by considering the ° terms of the right-hand member of equation (XXII) separately, and then adding them geometrically. It is evident from the curve that the value of. B starts at zero,                      FuL I4r>-
increases to a maximum, and then diminishes to zero asymptotically. In this case the motion is said to be aperiodic or dead-beat.
Case III.   Let_o^<62 then Va» V - i = i and Vb* - a2 =
this substitution in equation (XXI) we obtain
fl'-2i sin
/- is imaginary.    Let ~              Making
Sec Appendix Avu.